Chapter 13 Universal Gravitation
Jan 01, 2016
Chapter 13
Universal Gravitation
Newton’s Law of Universal Gravitation Every particle in the Universe attracts
every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the distance between them
G is the universal gravitational constant and equals 6.673 x 10-11 Nm2 / kg2
1 22g
mmF G
r
Law of Gravitation, cont This is an example of an inverse
square law The magnitude of the force varies as
the inverse square of the separation of the particles
The law can also be expressed in vector form
1 212 122
ˆmm
Gr
F r
Notation F12 is the force exerted by particle 1
on particle 2 The negative sign in the vector form
of the equation indicates that particle 2 is attracted toward particle 1
F21 is the force exerted by particle 2 on particle 1
More About Forces F12 = -F21
The forces form a Newton’s Third Law action-reaction pair
Gravitation is a field force that always exists between two particles, regardless of the medium between them
The force decreases rapidly as distance increases
A consequence of the inverse square law
G vs. g Always distinguish between G and g G is the universal gravitational
constant It is the same everywhere
g is the acceleration due to gravity g = 9.80 m/s2 at the surface of the Earth g will vary by location
Gravitational Force Due to a Distribution of Mass The gravitational force exerted by
a finite-size, spherically symmetric mass distribution on a particle outside the distribution is the same as if the entire mass of the distribution were concentrated at the center
For the Earth, 2E
gE
M mF G
R
Newton’s Verification He compared the acceleration of the
Moon in its orbit with the acceleration of an object falling near the Earth’s surface
He calculated the centripetal acceleration of the Moon from its distance and period
The high degree of agreement between the two techniques provided evidence of the inverse square nature of the law
Moon’s Acceleration Newton looked at proportionality of
accelerations between the Moon and objects on the Earth
2
2
2
1
1
MM E
M
E
ra R
g r
R
Centripetal Acceleration The Moon
experiences a centripetal acceleration as it orbits the Earth
22
2
2
2 /
4
MM
M M
M
r Tva
r r
r
T
Newton’s Assumption Newton treated the Earth as if its
mass were all concentrated at its center He found this very troubling
When he developed calculus, he showed this assumption was a natural consequence of the Law of Universal Gravitation
Measuring G G was first measured
by Henry Cavendish in 1798
The apparatus shown here allowed the attractive force between two spheres to cause the rod to rotate
The mirror amplifies the motion
It was repeated for various masses
Finding g from G The magnitude of the force acting on
an object of mass m in freefall near the Earth’s surface is mg
This can be set equal to the force of universal gravitation acting on the object
2
2
E
E
E
E
M mmg G
R
Mg G
R
g Above the Earth’s Surface If an object is some distance h above
the Earth’s surface, r becomes RE + h
This shows that g decreases with increasing altitude
As r , the weight of the object approaches zero
2E
E
GMg
R h
Variation of g with Height
Kepler’s Laws, Introduction Johannes Kepler was a German
astronomer He was Tycho Brahe’s assistant
Brahe was the last of the “naked eye” astronomers
Kepler analyzed Brahe’s data and formulated three laws of planetary motion
Kepler’s Laws Kepler’s First Law
All planets move in elliptical orbits with the Sun at one focus
Kepler’s Second Law The radius vector drawn from the Sun to a
planet sweeps out equal areas in equal time intervals
Kepler’s Third Law The square of the orbital period of any planet
is proportional to the cube of the semimajor axis of the elliptical orbit
Notes About Ellipses F1 and F2 are each a
focus of the ellipse They are located a
distance c from the center
The longest distance through the center is the major axis a is the semimajor
axis
Notes About Ellipses, cont The shortest distance
through the center is the minor axis b is the semiminor
axis The eccentricity of
the ellipse is defined as e = c /a For a circle, e = 0 The range of values of
the eccentricity for ellipses is 0 < e < 1
Notes About Ellipses, Planet Orbits The Sun is at one focus
Nothing is located at the other focus Aphelion is the point farthest away
from the Sun The distance for aphelion is a + c
For an orbit around the Earth, this point is called the apogee
Perihelion is the point nearest the Sun The distance for perihelion is a – c
For an orbit around the Earth, this point is called the perigee
Kepler’s First Law A circular orbit is a special case of the
general elliptical orbits Is a direct result of the inverse square
nature of the gravitational force Elliptical (and circular) orbits are allowed
for bound objects A bound object repeatedly orbits the center An unbound object would pass by and not
return These objects could have paths that are parabolas (e = 1) and hyperbolas (e > 1)
Orbit Examples Pluto has the
highest eccentricity of any planet (a) ePluto = 0.25
Halley’s comet has an orbit with high eccentricity (b) eHalley’s comet = 0.97
Kepler’s Second Law Is a consequence of
conservation of angular momentum
The force produces no torque, so angular momentum is conserved
L = r x p = MP r x v = 0
Kepler’s Second Law, cont. Geometrically, in a
time dt, the radius vector r sweeps out the area dA, which is half the area of the parallelogram
|r x dr| Its displacement is
given by d r = v dt
Kepler’s Second Law, final Mathematically, we can say
The radius vector from the Sun to any planet sweeps out equal areas in equal times
The law applies to any central force, whether inverse-square or not
constant2 p
dA L
dt M
Kepler’s Third Law Can be predicted
from the inverse square law
Start by assuming a circular orbit
The gravitational force supplies a centripetal force
Ks is a constant
2Sun Planet Planet
2
22 3 3
Sun
2
4S
GM M M v
r rr
vT
T r K rGM
Kepler’s Third Law, cont This can be extended to an elliptical
orbit Replace r with a
Remember a is the semimajor axis
Ks is independent of the mass of the planet, and so is valid for any planet
22 3 3
Sun
4ST a K a
GM
Kepler’s Third Law, final If an object is orbiting another
object, the value of K will depend on the object being orbited
For example, for the Moon around the Earth, KSun is replaced with KEarth
Example, Mass of the Sun Using the distance between the Earth
and the Sun, and the period of the Earth’s orbit, Kepler’s Third Law can be used to find the mass of the Sun
Similarly, the mass of any object being orbited can be found if you know information about objects orbiting it
2 3
Sun 2
4 rM
GT
Example, Geosynchronous Satellite A geosynchronous
satellite appears to remain over the same point on the Earth
The gravitational force supplies a centripetal force
You can find h or v
The Gravitational Field A gravitational field exists at
every point in space When a particle of mass m is
placed at a point where the gravitational field is g, the particle experiences a force Fg = m g
The field exerts a force on the particle
The Gravitational Field, 2 The gravitational field g is defined as
The gravitational field is the gravitational force experienced by a test particle placed at that point divided by the mass of the test particle
The presence of the test particle is not necessary for the field to exist
g
mF
g
The Gravitational Field, 3 The gravitational
field vectors point in the direction of the acceleration a particle would experience if placed in that field
The magnitude is that of the freefall acceleration at that location
The Gravitational Field, final The gravitational field describes
the “effect” that any object has on the empty space around itself in terms of the force that would be present if a second object were somewhere in that space
2ˆg GM
m r F
g r
Gravitational Potential Energy The gravitational force is
conservative The gravitational force is a central
force It is directed along a radial line
toward the center Its magnitude depends only on r A central force can be represented by
ˆF r r
Grav. Potential Energy – Work A particle moves from
A to B while acted on by a central force F
The path is broken into a series of radial segments and arcs
Because the work done along the arcs is zero, the work done is independent of the path and depends only on rf and ri
Grav. Potential Energy – Work, cont The work done by F along any radial
segment is
The work done by a force that is perpendicular to the displacement is 0
The total work is
Therefore, the work is independent of the path
( )dW d F r dr F r
( ) f
i
r
rW F r dr
Gravitational Potential Energy, cont As a particle moves
from A to B, its gravitational potential energy changes by
This is the general form, we need to look at gravitational force specifically
( ) f
i
r
f i rU U U F r dr
Gravitational Potential Energy for the Earth Choose the zero for the gravitational
potential energy where the force is zero This means Ui = 0 where ri = ∞
This is valid only for r≥RE and not valid for r < RE
U is negative because of the choice of Ui
( ) EGM mU r
r
Gravitational Potential Energy for the Earth, cont Graph of the
gravitational potential energy U versus r for an object above the Earth’s surface
The potential energy goes to zero as r approaches infinity
Gravitational Potential Energy, General For any two particles, the gravitational
potential energy function becomes
The gravitational potential energy between any two particles varies as 1/r Remember the force varies as 1/r 2
The potential energy is negative because the force is attractive and we chose the potential energy to be zero at infinite separation
1 2GmmU
r
Gravitational Potential Energy, General cont An external agent must do positive
work to increase the separation between two objects The work done by the external agent
produces an increase in the gravitational potential energy as the particles are separated
U becomes less negative
Binding Energy The absolute value of the potential
energy can be thought of as the binding energy
If an external agent applies a force larger than the binding energy, the excess energy will be in the form of kinetic energy of the particles when they are at infinite separation
Systems with Three or More Particles
The total gravitational potential energy of the system is the sum over all pairs of particles
Gravitational potential energy obeys the superposition principle
Systems with Three or More Particles, cont Each pair of particles contributes a term
of U Assuming three particles:
The absolute value of Utotal represents the work needed to separate the particles by an infinite distance
total 12 13 23
1 2 1 3 2 3
12 13 23
U U U U
mm mm m mG
r r r
Energy and Satellite Motion Assume an object of mass m moving
with a speed v in the vicinity of a massive object of mass M M >>m
Also assume M is at rest in an inertial reference frame
The total energy is the sum of the system’s kinetic and potential energies
Energy and Satellite Motion, 2 Total energy E = K +U
In a bound system, E is necessarily less than 0
21
2
MmE mv G
r
Energy in a Circular Orbit An object of mass
m is moving in a circular orbit about M
The gravitational force supplies a centripetal force
2
GMmE
r
Energy in a Circular Orbit, cont The total mechanical energy is
negative in the case of a circular orbit
The kinetic energy is positive and is equal to half the absolute value of the potential energy
The absolute value of E is equal to the binding energy of the system
Energy in an Elliptical Orbit For an elliptical orbit, the radius is
replaced by the semimajor axis
The total mechanical energy is negative
The total energy is constant if the system is isolated
2
GMmE
a
Summary of Two Particle Bound System Total energy is
Both the total energy and the total angular momentum of a gravitationally bound, two-object system are constants of the motion
2 21 1
2 2i fi f
GMm GMmE mv mv
r r
Escape Speed from Earth An object of mass m is
projected upward from the Earth’s surface with an initial speed, vi
Use energy considerations to find the minimum value of the initial speed needed to allow the object to move infinitely far away from the Earth
Escape Speed From Earth, cont This minimum speed is called the
escape speed
Note, vesc is independent of the mass of the object
The result is independent of the direction of the velocity and ignores air resistance
2 Eesc
E
GMv
R
Escape Speed, General The Earth’s result
can be extended to any planet
The table at right gives some escape speeds from various objects
esc
2GMv
R
Escape Speed, Implications Complete escape from an object is
not really possible The gravitational field is infinite and so
some gravitational force will always be felt no matter how far away you can get
This explains why some planets have atmospheres and others do not Lighter molecules have higher average
speeds and are more likely to reach escape speeds
Black Holes A black hole is the remains of a
star that has collapsed under its own gravitational force
The escape speed for a black hole is very large due to the concentration of a large mass into a sphere of very small radius If the escape speed exceeds the
speed of light, radiation cannot escape and it appears black
Black Holes, cont The critical radius at
which the escape speed equals c is called the Schwarzschild radius, RS
The imaginary surface of a sphere with this radius is called the event horizon
This is the limit of how close you can approach the black hole and still escape
Black Holes and Accretion Disks Although light from a black hole
cannot escape, light from events taking place near the black hole should be visible
If a binary star system has a black hole and a normal star, the material from the normal star can be pulled into the black hole
Black Holes and Accretion Disks, cont This material forms
an accretion disk around the black hole
Friction among the particles in the disk transforms mechanical energy into internal energy
Black Holes and Accretion Disks, final The orbital height of the material
above the event horizon decreases and the temperature rises
The high-temperature material emits radiation, extending well into the x-ray region
These x-rays are characteristics of black holes
Black Holes at Centers of Galaxies There is evidence
that supermassive black holes exist at the centers of galaxies
Theory predicts jets of materials should be evident along the rotational axis of the black hole
An HST image of the galaxy M87. The jet of material in the right frame is thought to be evidence of a supermassive black hole at the galaxy’s center.